AS PURE MATHS REVISION BOOKLEI
Quadratics
tb-42
To sove a" +hr+c =0, factorise, use the formula r =- or cakculalor.
2a
real roots depanding whether
The discriminant is
-4ac There will
be wo distinct roots, equal roots
the discnminant is positive, zero or negative.
or no on
Completing the square -replace x+bx with ( x + - ( orx -bx with ( x - ) - G
Graph of y = ax
+bx +cis aparabolawith U shape when a>0, and n shape when
The graph crosses the y axis when x = 0, s0y C
a < 0.
The graph crosses the x axis wheny = 0, so sove ar + bx t e =0.
When In completed square form, the vertex of y=a(x+ p) +q has coordinates (-p.9), the equation of the
ine of symmetry isX-p
For intervals of the number ine, (a, b) ={xe R:a<1< b}; la,b])= {xe R:asIsb
o sove linear inegualites, treal as equations, but if you divide or multiply by a negalive, reverSe the sign.
» 0 when the graph is above the xaxis,
Tosolve aquadraticinequality, sketchthe graphofthequadralic -if's
R s O W n e nthe graph is below the x axis.
Epress r-8+ 1gin the form (-pY* 9. herep and q are inlegers.
)ence. or otherMse. show that the equation - 8r + 19 - 0 h a s no real so ulions
Skech te graph ofy--8 19. stating the coordinates af the mnimumpoint and the
point where the graph erosses the - a s
Find the values of t for which the ecuation -2E-1)x-2t*-7-0 has equal roots
3)Factorse9-8r-
b) Show that
25-r 4r
can be w t e n as
9-8
-
9 -8r-*
A Gune has equatonr
Wrte down the equation of ts hne of symmmelry.
F i n d the coordinales of fs vertex
) Sketch the curve, indicating the vaues of the interoeps on the x-axs and the -ads.
ind the values for x which saisty both inequalites r 2 r > 8 and 3{2r-1)s15.
Surds,Indices and Equations
To simplity surds, use yab vavb and
To ratonalise the denominator of e. g multiply top & bottom byy2.
To rationalise the denorminator of e. 9 mutiply top & bottom by 4 + v3.
Rules of Indices:
a" " =am*n; am; («"m)" = am*; a - 1; «" = a = Va; a = (Va)".
For simultaneous equations, ane linear & one quadratic, rearrange the linear one to make x or y the subject.
substtute into the
quadralic, simplily & solve.
intersection of curves, solve If line is a tangent to a curve, the equalion
To find the points of simultaneously. a
oained by solving them simullaneously wil have egual rools.
Disguised quadratics: . g. 2"" + 32-0;et y so 2y43y-2-0.
Solve to Iind y & henca x.
, Exampless
a) Wrde dow the valuesofp.q and r. gven tha
54
b F n d the value
of *r for which
Show that i s an inleger and find ts value
3
A rectangte has length 19-53jcmand ares (15-73 |cm
Shew
tha: the width of the rectangie is (-n3).where m and n are integers
t a s i n g the siubsituton 3. show that the equation
9-3''-2 0
can be witen as
T-1-2)-0
br Hence showthat the aqualion
-3 "2-0
has a solution
and. by using logarithms, find the other soluion grving your answer to four decma
places.
A arde wth centre Chas equation
--10y 20-0
A line has equation
a) Show that the -coordinate of any point of mtersecion at the ime and the arce satisfes
the equationx-Ar-4-0
0 e n e o w that the ine is a tangent to tne circle and find the coorcinates of the point of
The curve Chas equalion
tr-3)
e r e ksa constant
The ine L has equation
-2r+2
how hat the x-cnordienates of any points of ntersection of the curve Cwih the ine L satisly
the equaton
k - 2 r 3k-2-D
Quadratics
tb-42
To sove a" +hr+c =0, factorise, use the formula r =- or cakculalor.
2a
real roots depanding whether
The discriminant is
-4ac There will
be wo distinct roots, equal roots
the discnminant is positive, zero or negative.
or no on
Completing the square -replace x+bx with ( x + - ( orx -bx with ( x - ) - G
Graph of y = ax
+bx +cis aparabolawith U shape when a>0, and n shape when
The graph crosses the y axis when x = 0, s0y C
a < 0.
The graph crosses the x axis wheny = 0, so sove ar + bx t e =0.
When In completed square form, the vertex of y=a(x+ p) +q has coordinates (-p.9), the equation of the
ine of symmetry isX-p
For intervals of the number ine, (a, b) ={xe R:a<1< b}; la,b])= {xe R:asIsb
o sove linear inegualites, treal as equations, but if you divide or multiply by a negalive, reverSe the sign.
» 0 when the graph is above the xaxis,
Tosolve aquadraticinequality, sketchthe graphofthequadralic -if's
R s O W n e nthe graph is below the x axis.
Epress r-8+ 1gin the form (-pY* 9. herep and q are inlegers.
)ence. or otherMse. show that the equation - 8r + 19 - 0 h a s no real so ulions
Skech te graph ofy--8 19. stating the coordinates af the mnimumpoint and the
point where the graph erosses the - a s
Find the values of t for which the ecuation -2E-1)x-2t*-7-0 has equal roots
3)Factorse9-8r-
b) Show that
25-r 4r
can be w t e n as
9-8
-
9 -8r-*
A Gune has equatonr
Wrte down the equation of ts hne of symmmelry.
F i n d the coordinales of fs vertex
) Sketch the curve, indicating the vaues of the interoeps on the x-axs and the -ads.
ind the values for x which saisty both inequalites r 2 r > 8 and 3{2r-1)s15.
Surds,Indices and Equations
To simplity surds, use yab vavb and
To ratonalise the denominator of e. g multiply top & bottom byy2.
To rationalise the denorminator of e. 9 mutiply top & bottom by 4 + v3.
Rules of Indices:
a" " =am*n; am; («"m)" = am*; a - 1; «" = a = Va; a = (Va)".
For simultaneous equations, ane linear & one quadratic, rearrange the linear one to make x or y the subject.
substtute into the
quadralic, simplily & solve.
intersection of curves, solve If line is a tangent to a curve, the equalion
To find the points of simultaneously. a
oained by solving them simullaneously wil have egual rools.
Disguised quadratics: . g. 2"" + 32-0;et y so 2y43y-2-0.
Solve to Iind y & henca x.
, Exampless
a) Wrde dow the valuesofp.q and r. gven tha
54
b F n d the value
of *r for which
Show that i s an inleger and find ts value
3
A rectangte has length 19-53jcmand ares (15-73 |cm
Shew
tha: the width of the rectangie is (-n3).where m and n are integers
t a s i n g the siubsituton 3. show that the equation
9-3''-2 0
can be witen as
T-1-2)-0
br Hence showthat the aqualion
-3 "2-0
has a solution
and. by using logarithms, find the other soluion grving your answer to four decma
places.
A arde wth centre Chas equation
--10y 20-0
A line has equation
a) Show that the -coordinate of any point of mtersecion at the ime and the arce satisfes
the equationx-Ar-4-0
0 e n e o w that the ine is a tangent to tne circle and find the coorcinates of the point of
The curve Chas equalion
tr-3)
e r e ksa constant
The ine L has equation
-2r+2
how hat the x-cnordienates of any points of ntersection of the curve Cwih the ine L satisly
the equaton
k - 2 r 3k-2-D
